According to the U.S. Census Bureau, 7.1 % of all babies...

According to the U.S. Census Bureau, $7.1 \%$ of all babies born to nonsmoking mothers are of low birth weight $(<51 b, 8$ oz ). An obstetrician wanted to know whether mothers between the ages of 35 and 39 years give birth to a higher percentage of low-birth-weight babies. She randomly selected 160 births for which the mother was 35 to 39 years old and found 15 low-birth-weight babies. (a) If the proportion of low-birth-weight babies for mothers in this age group is $0.071,$ compute the expected number of low-birth-weight births to 35 - to 39 -year old mothers. What is the expected number of births to mothers 35 to 39 years old that are not low birth weight? (b) Test the hypothesis at the $\alpha=0.05$ level of significance using the chi-square goodness-of-fit test. (c) Test the hypothesis using the approach presented in Section 10.4.

According to the U.S. Census Bureau, $7.1 \%$ of all babies born to nonsmoking mothers are of low birth weight $(<51 b, 8$ oz ). An obstetrician wanted to know whether mothers between the ages of 35 and 39 years give birth to a higher percentage of low-birth-weight babies. She randomly selected 160 births for which the mother was
35 to 39 years old and found 15 low-birth-weight babies.
(a) If the proportion of low-birth-weight babies for mothers in this age group is $0.071,$ compute the expected number of low-birth-weight births to 35 - to 39 -year old mothers. What is the expected number of births to mothers 35 to 39 years old that are not low birth weight?
(b) Test the hypothesis at the $\alpha=0.05$ level of significance using the chi-square goodness-of-fit test.
(c) Test the hypothesis using the approach presented in Section 10.4.

Key Concepts

Expected Frequency Calculation

This concept involves determining the number of cases expected in each category under a given hypothesis. In proportion testing, the expected frequency is calculated by multiplying the total number of observations by the hypothesized proportion. This calculation is fundamental when comparing observed data to what is anticipated under a null hypothesis.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical method used to evaluate how well observed categorical data match expected frequencies derived from a specified theoretical distribution. The test statistic is computed by summing the squared differences between observed and expected frequencies divided by the expected frequencies, and the result is compared against a chi-square distribution with the appropriate degrees of freedom to determine statistical significance.

Hypothesis Testing Framework

This concept outlines the process of making statistical decisions based on sample data. It involves setting up a null hypothesis that represents the status quo (e.g., the proportion of low-birth-weight births is 7.1%) and an alternative hypothesis suggesting deviation from the null. A significance level (such as 0.05) is predetermined to control the probability of a Type I error, guiding whether to reject the null hypothesis based on the evidence.

Normal Approximation for Proportions

When dealing with large sample sizes, the sampling distribution of a proportion can be approximated by a normal distribution. This allows for the use of z-tests to assess differences between the observed sample proportion and the hypothesized population proportion. The normal approximation is a powerful tool for conducting hypothesis tests when the conditions for applying the central limit theorem are met.

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